In this article ideas from Kit Fine’s theory of arbitrary objects are applied to questions regarding mathematical structuralism. I discuss how sui generis mathematical structures can be viewed as generic systems of mathematical objects, where mathematical objects are conceived of as arbitrary objects in Fine’s sense.

'Chains of Being' argues that there can be infinite chains of dependence or grounding. Cameron also defends the view that there can be circular relations of ontological dependence or grounding, and uses these claims to explore issues in logic and ontology.

The paper challenges Williamson’s safety based explanation for why we cannot know the cut-off point of vague expressions. We assume throughout (most of) the paper that Williamson is correct in saying that vague expressions have sharp cut-off points, but we argue that Williamson’s explanation for why we do not and cannot know these cut-off points is unsatisfactory. -/- In sect 2 we present Williamson's position in some detail. In particular, we note that Williamson's explanation relies on taking a particular safety (...) principle ('Meta-linguistic belief safety' or 'MBS') as a necessary condition on knowledge. In section 3, we show that even if MBS were a necessary condition on knowledge, that would not be sufficient to show that we cannot know the cut-off points of vague expressions. In section 4, we present our main case against Williamson's explanation: we argue that MBS is not a necessary condition on knowledge, by presenting a series of cases where one's belief violates MBS but nevertheless constitutes knowledge. In section 5, we present and respond to an objection to our view. And in section 6, we briefly discuss the possible directions a theory of vagueness can take, if our objection to Williamson's theory is taken on board. (shrink)

Presenting the first comprehensive, in-depth study of hyperintensionality, this book equips readers with the basic tools needed to appreciate some of current and future debates in the philosophy of language, semantics, and metaphysics. After introducing and explaining the major approaches to hyperintensionality found in the literature, the book tackles its systematic connections to normativity and offers some contributions to the current debates. The book offers undergraduate and graduate students an essential introduction to the topic, while also helping professionals in related (...) fields get up to speed on open research-level problems. (shrink)

The aim of this paper is to introduce, elucidate and defend the usefulness of a variant of grounding, or metaphysical explanation, that has the feature that the grounds explain of some states of affairs that one of them obtains without explaining which one obtains. I will dub this variant arbitrary grounding. After informally elucidating the basic idea in the first section, I will provide three metaphysical hypotheses that are best formulated in terms of arbitrary grounding in the second section. The (...) third section will be concerned with the relation between arbitrary grounding and non-arbitrary grounding. The fourth section will compare arbitrary grounding to two extant proposals in the literature. (shrink)

This paper defends stage theory against the argument from diachronic counting. It argues that stage theorists can appeal to quantifier domain restriction in order to accommodate intuitions about diachronic counting sentences. Two approaches involving domain restriction are discussed. According to the first, domains of counting are usually restricted to stages at the time of utterance. This approach explains intuitions in many cases, but is theoretically costly and delivers wrong counts if diachronic counting is combined with fission or fusion. On the (...) second approach, domains of counting are usually restricted in an indeterminate way, so as to include at most one member of any maximal class of counterpart-interrelated stages (with respect to a certain utterance). This view can accommodate all the relevant intuitions about counting sentences, and it fits well with a new stage-theoretic view of reference that allows speakers to refer to both present and past stages. (shrink)

This paper concerns the apparent fact — discussed by Sinan Dogramaci and Brian Weatherson — that inductive reasoning often interacts in disastrous ways with patterns of reasoning that seem perfectly fine in the deductive case. In contrast to Dogramaci's and Weatherson's own suggestions, I argue that these cases show that we cannot reason inductively about arbitrary objects. Moreover, as I argue, this prohibition is neatly explained by a certain hypothesis about the rational basis of inductive reasoning — namely, the hypothesis (...) that inductive reasoning is fundamentally reasoning about what normally happens. (shrink)

This paper concerns the apparent fact — discussed by Sinan Dogramaci and Brian Weatherson — that inductive reasoning often interacts in disastrous ways with patterns of reasoning that seem perfectly fine in the deductive case. In contrast to Dogramaci's and Weatherson's own suggestions, I argue that these cases show that we cannot reason inductively about arbitrary objects. Moreover, as I argue, this prohibition is neatly explained by a certain hypothesis about the rational basis of inductive reasoning — namely, the hypothesis (...) that inductive reasoning is fundamentally reasoning about what normally happens. (shrink)

‘Ahab is a whaler’ and ‘Holmes is a whaler’ express different propositions, even though neither ‘Ahab’ nor ‘Holmes’ has a referent. This seems to constitute a theoretical puzzle for the Russellian view of propositions. In this paper, I develop a variant of the Russellian view, Plenitudinous Russellianism. I claim that ‘Ahab is a whaler’ and ‘Holmes is a whaler’ express distinct gappy propositions. I discuss key metaphysical and semantic differences between Plenitudinous Russellianism and Traditional Russellianism and respond to objections that (...) stem from those differences. (shrink)

There is an interesting logical/semantic issue with some mathematical languages and theories. In the language of (pure) complex analysis, the two square roots of i’ manage to pick out a unique object? This is perhaps the most prominent example of the phenomenon, but there are some others. The issue is related to matters concerning the use of definite descriptions and singular pronouns, such as donkey anaphora and the problem of indistinguishable participants. Taking a cue from some work in linguistics and (...) the philosophy of language, I suggest that i functions like a parameter in natural deduction systems. This may require some rethinking of the role of singular terms, at least in mathematical languages. (shrink)

Rudolf Carnap’s mature work on the logical reconstruction of scientific theories consists of two components. The first is the elimination of the theoretical vocabulary of a theory in terms of its Ramsification. The second is the reintroduction of the theoretical terms through explicit definitions in a language containing an epsilon operator. This paper investigates Carnap’s epsilon-reconstruction of theories in the context of pure mathematics. The main objective here is twofold: first, to specify the epsilon logic underlying his suggested definition of (...) theoretical terms and a suitable choice semantics for it. Second, to analyze whether Carnap’s approach is compatible with a structuralist conception of mathematics. (shrink)

The aim of this essay is to argue that, if a robust form of moral realism is true, then moral vagueness is ontic vagueness. The argument is by elimination: I show that neither semantic nor epistemic approaches to moral vagueness are satisfactory.

The pattern of credences we are inclined to assign to counterfactuals challenges standard accounts of counterfactuals. In response to this problem, the paper develops a semantics of counterfactuals in terms of the epsilon-operator. The proposed semantics stays close to the standard account: the epsilon-operator substitutes the universal quantifier present in standard semantics by arbitrarily binding the open world-variable. Various applications of the suggested semantics are explored including, in particular, an explanation of how the puzzling credences in counterfactuals come about.

In the natural deduction system for classical propositional logic given by G. Gentzen, there are some inference rules with assumptions discharged by the rule. D. Prawitz calls such inference rules improper, and others proper. Improper inference rules are more complicated and are often harder to understand than the proper ones. In the present paper, we distinguish between proper and improper derivations by using sequent systems. Specifically, we introduce a sequent system \(\vdash_{\bf Sc}\) for classical propositional logic with only structural rules, (...) and prove that \(\vdash_{\bf Sc}\) does not allow improper derivations in general. For instance, the sequent \(\Rightarrow p \to q\) cannot be derived from the sequent \(p \Rightarrow q\) in \(\vdash_{\bf Sc}\). In order to prove the failure of improper derivations, we modify the usual notion of truth valuation, and using the modified valuation, we prove the completeness of \(\vdash_{\bf Sc}\). We also consider whether an improper derivation can be described generally by using \(\vdash_{\bf Sc}\). (shrink)

In the natural deduction system for classical propositional logic given by G. Gentzen, there are some inference rules with assumptions discharged by the rule. D. Prawitz calls such inference rules improper as opposed to proper ones. Improper inference rules are more complicated than proper ones and more difficult to understand. In 2022, we provided a sequent system based solely on the application of proper rules. In the present paper, on the basis of our system from 2022, we classify improper inference (...) rules. (shrink)

Traditional procedures for rational updating fail when it comes to self-locating opinions, such as your credences about where you are and what time it is. This paper develops an updating procedure for rational agents with self-locating beliefs. In short, I argue that rational updating can be factored into two steps. The first step uses information you recall from your previous self to form a hypothetical credence distribution, and the second step changes this hypothetical distribution to reflect information you have genuinely (...) learned as time has passed. While the second step resembles traditional procedures of updating by conditionalization, the first is best understood by analogy to traditional models of how agents transmit self-locating opinions through ordinary interpersonal communication. (shrink)

Reductionist realist accounts of certain entities, such as the natural numbers and propositions, have been taken to be fatally undermined by what we may call the problem of arbitrary identification. The problem is that there are multiple and equally adequate reductions of the natural numbers to sets (see Benacerraf, 1965), as well as of propositions to unstructured or structured entities (see, e.g., Bealer, 1998; King, Soames, & Speaks, 2014; Melia, 1992). This paper sets out to solve the problem by canvassing (...) what we may call the arbitrary reference strategy. The main claims of such strategy are 2. First, we do not know which objects are the referents of proposition and numerical terms since their reference is fixed arbitrarily. Second, our ignorance of which object is picked out as the referent does not entail that no object is referred to by the relevant expression. Different articulations of the strategy are assessed, and a new one is defended. (shrink)

The notorious problem of the many makes it difficult to resist the conclusion that almost coincident with any ordinary object are a vast number of near-indiscernible objects. As Unger was aware in his presentation of the problem, this abundance raises a concern as to how—and even whether—we achieve singular thought about ordinary objects. This paper presents, clarifies, and defends a view which reconciles a plenitudinous conception of ordinary objects with our having singular thoughts about those objects. Indeed, this strategy has (...) independent application in the case of singular thoughts about other putatively ‘abundant’ phenomena, such as locations or lumps of matter. In essence, singular thought-vehicles need not express just one singular content. If there are many objects, one’s singular thought-vehicle may express as many thought-contents. (shrink)

This paper argues that several leading theories of subjunctive conditionals are incompatible with ordinary intuitions about what credences we ought to have in subjunctive conditionals. In short, our theory of subjunctives should intuitively display semantic humility, i.e. our semantic theory should deliver the truth conditions of sentences without pronouncing on whether those conditions actually obtain. In addition to describing intuitions about subjunctive conditionals, I argue that we can derive these ordinary intuitions from justified premises, and I answer a possible worry (...) for my derivation by refuting a subjunctive triviality result modeled on (Lewis 1976). I conclude that the debate over the correct theory of subjunctive conditionals requires settling meta-philosophical questions about the relative value of various virtues of first-order theories of subjunctive conditionals. (shrink)

This paper is a response to Haze’s brief argument for the falsity of the theory that instantial terms refer arbitrarily, proposed by Breckenridge and Magidor in 2012. In this paper, I characterise instantial terms and outline the theory of arbitrary reference; then I reconstruct Haze’s argument and contend that it fails in its purpose. Haze’s argument is supposed to be a _reductio ad absurdum:_ according to Haze, it proves that a contradiction follows from the most basic tenets of the theory (...) of arbitrary reference. I will argue, however, that the contradiction in question follows not from these tenets, but from the surreptitious use that Haze makes of a self-referential expression. I conclude, consequently, that Haze’s argument is nothing more than an illustration of the well-known fact that self-referential expressions produce paradoxical results. (shrink)

Is it possible to maintain classical logic, stay close to classical semantics, and yet accept that language might be semantically indeterminate? The article gives an affirmative answer by Ramsifying classical semantics, which yields a new semantic theory that remains much closer to classical semantics than supervaluationism but which at the same time avoids the problematic classical presupposition of semantic determinacy. The resulting Ramsey semantics is developed in detail, it is shown to supply a classical concept of truth and to fully (...) support the rules and metarules of classical logic, and it is applied to vague terms as well as to theoretical or open-ended terms from mathematics and science. The theory also demonstrates how diachronic or synchronic interpretational continuity across languages is compatible with semantic indeterminacy. (shrink)

We offer an interpretation of the words and works of Richard Dedekind and the David Hilbert of around 1900 on which they are held to entertain diverging views on the structure of a deductive science. Firstly, it is argued that Dedekind sees the beginnings of a science in concepts, whereas Hilbert sees such beginnings in axioms. Secondly, it is argued that for Dedekind, the primitive terms of a science are substantive terms whose sense is to be conveyed by elucidation, whereas (...) Hilbert dismisses elucidation and consequently treats the primitives as schematic. (shrink)

I know that I could have been where you are right now and that you could have been where I am right now, but that neither of us could have been turnips or natural numbers. This knowledge of metaphysical modality stands in need of explanation. I will offer an account based on our knowledge of the natures, or essencess, of things. I will argue that essences need not be viewed as metaphysically bizarre entities; that we can conceptualise and refer to (...) essences; and that we can gain knowledge of them. We can know about which properties are, and which properties are not, essential to a given entity. This knowledge of essence offers a route to knowledge of the ways those entities must be or could be. (shrink)

I know that I could have been where you are right now and that you could have been where I am right now, but that neither of us could have been turnips or natural numbers. This knowledge of metaphysical modality stands in need of explanation. I will offer an account based on our knowledge of the natures, or essencess, of things. I will argue that essences need not be viewed as metaphysically bizarre entities; that we can conceptualise and refer to (...) essences; and that we can gain knowledge of them. We can know about which properties are, and which properties are not, essential to a given entity. This knowledge of essence offers a route to knowledge of the ways those entities must be or could be. (shrink)

The Received View of particles in quantum mechanics is that they are indistinguishable entities within their kinds and that, as a consequence, they are not individuals in the metaphysical sense and self-identity does not meaningfully apply to them. Nevertheless cardinality does apply, in that one can have n> 1 such particles. A number of authors have recently argued that this cluster of claims is internally contradictory: roughly, that having more than one such particle requires that the concepts of distinctness and (...) identity must apply after all. A common thread here is that the notion of identity is too fundamental to forego in any metaphysical account. I argue that this argument fails. I then argue that the failure of individuality and identity applies also to macroscopic physical objects, that the problems cannot be constrained to apply only within the microscopic realm. (shrink)

In a recent paper, Breckenridge and Magidor argue for an interesting and counterintuitive account of instantial reasoning. According to this account, in arguments such as one beginning with 'There is some x such that x is mortal. Let O be such an x. ...', the 'O' refers to a particular object, although we cannot know which. I give and defend a simple counterexample involving the notion of an unreferred-to object.

Non-actualist theories promise straightforward accounts of meaning, truth and reference of fictional discourse but are ostensibly saddled with a Selection Problem, that multiple possible candidates satisfy the role-descriptions associated with names used in fictions and no principled way to distinguish between them; yet if names are referential, there can only be one referent. The problem is often taken to be a serious—even decisive—obstacle for non-actualism, and the aim of this article is to show that the challenge can be met. I (...) suggest that storytellers and authors fix the referents of referential terms they use by arbitrary selecting referents through simple acts of stipulation, and then determine which of the worlds containing the relevant individuals serve as truth-makers for the story by adding properties to the characters. The resulting view of reference is consistent with reasonable, foundational requirements on reference, such as a causal-historical theory, and can moreover be used to support plausible theories of truth in fiction and facilitate elegant models of fiction as a type of discourse. (shrink)

Ante rem structures were posited as the subject matter of mathematics in order to resolve a problem of referential indeterminacy within mathematical discourse. Nevertheless, ante rem structuralists are inevitably committed to the existence of indiscernible entities, and this commitment produces an exactly analogous problem. If it cannot be sorted out, then the postulation of ante rem structures is futile. In a recent paper, Stewart Shapiro argued that the problem may be solved by analysing some of the singular terms of mathematics (...) not as genuinely referring expressions, but as instantial terms. In this paper, I discuss several competing accounts of the semantics of terms of this kind, and argue that they are all untenable for the ante rem structuralist. Shapiro, then, still owes us an account of the semantics of instantial terms that suits the ante rem structuralist project. Without it, the ante rem structuralist is still unable to determine the reference of the singular terms of mathematics. (shrink)

This paper investigates ‘authoritative knowledge’, a neglected species of practical knowledge gained on the basis of exercising practical authority. I argue that, like perceptual knowledge, authoritative knowledge is non-inferential. I then present a broadly reliabilist account of the process by which authority yields knowledge, and use this account to address certain objections.

Arbitrariness is a distinctive feature of human language, and a growing body of comparative work is investigating its presence in animal communication. But what is arbitrariness, exactly? We propose to distinguish four notions of semiotic arbitrariness: a notion of opaque association between sign forms and semiotic functions, one of sign‐function mapping optionality, one of acquisition‐dependent sign‐function coupling, and one of lack of motivatedness. We characterize these notions, illustrate the benefits of keeping them apart, and describe two reactions to our proposal: (...) abandoning arbitrariness‐talk in favor of the newly introduced conceptual vocabulary, or feeding the distinctions back into the parent concept. (shrink)

Arbitrariness is a distinctive feature of human language, and a growing body of comparative work is investigating its presence in animal communication. But what is arbitrariness, exactly? We propose to distinguish four notions of semiotic arbitrariness: a notion of opaque association between sign forms and semiotic functions, one of sign-function mapping optionality, one of acquisition-dependent sign-function coupling, and one of lack of motivatedness. We characterize these notions, illustrate the benefits of keeping them apart, and describe two reactions to our proposal: (...) abandoning arbitrariness-talk in favor of the newly introduced conceptual vocabulary, or feeding the distinctions back into the parent concept. (shrink)

In To be is to be the object of a possible act of choice the authors defended Boolos’ thesis that plural quantification is part of logic. To this purpose, plural quantification was explained in terms of plural reference, and a semantics of plural acts of choice, performed by an ideal team of agents, was introduced. In this paper, following that approach, we develop a theory of concepts that—in a sense to be explained—can be labeled as a theory of logical concepts. (...) Within this theory, we propose a new logicist approach to natural numbers. Then, we compare our logicism with Frege’s traditional logicism. (shrink)

In an addendum to Naming and Necessity, Saul Kripke argues against his earlier view that Sherlock Holmes is a possible person. In this paper, I suggest a nonstandard interpretation of the addendum. A key feature of this non-standard interpretation is that it attempts to make sense of why Kripke would be rejecting the view that Sherlock Holmes is a possible person without asserting that it is not the case that Sherlock Holmes is a possible person.

It is argued that truth value of a sentence containing free variables in a context of use, just as the reference of the free variables concerned, depends on the assumptions and posits given by the context. However, context may under-determine the reference of a free variable and the truth value of sentences in which it occurs. It is argued that in such cases a free variable has indeterminate reference and a sentence in which it occurs may have indeterminate truth value. (...) On letting, say, x be such that \, the sentence ‘Either \ or \’ is true but the sentence ‘\’ has an indeterminate truth value: it is determinate that the variable x refers to either 2 or \, but it is indeterminate which of the two it refers to, as a result ‘\’ has a truth value but its truth value is indeterminate. The semantic indeterminacy is analysed in a ‘radically’ supervaluational semantic framework closely analogous to the treatment of vagueness in McGee and McLaughlin and Smith, which saves bivalence, the T-schema and the truth-functional analysis of the boolean connectives. It is shown that on such an analysis the modality ‘determinately’ is quite clearly not an epistemic modality, avoiding a potential objection raised by Williamson against such ‘radically’ supervaluational treatments of vagueness, and that determinate truth is the semantic value preserved in classically valid arguments. The analysis is contrasted with the epistemicist proposal of Breckenridge and Magidor which implies that ‘\’ has a determinate but unknowable truth value. (shrink)

Cantor’s abstractionist account of cardinal numbers has been criticized by Frege as a psychological theory of numbers which leads to contradiction. The aim of the paper is to meet these objections by proposing a reassessment of Cantor’s proposal based upon the set theoretic framework of Bourbaki—called BK—which is a First-order set theory extended with Hilbert’s \-operator. Moreover, it is argued that the BK system and the \-operator provide a faithful reconstruction of Cantor’s insights on cardinal numbers. I will introduce first (...) the axiomatic setting of BK and the definition of cardinal numbers by means of the \-operator. Then, after presenting Cantor’s abstractionist theory, I will point out two assumptions concerning the definition of cardinal numbers that are deeply rooted in Cantor’s work. I will claim that these assumptions are supported as well by the BK definition of cardinal numbers, which will be compared to those of Zermelo–von Neumann and Frege–Russell. On the basis of these similarities, I will make use of the BK framework in meeting Frege’s objections to Cantor’s proposal. A key ingredient in the defence of Cantorian abstraction will be played by the role of representative sets, which are arbitrarily denoted by the \-operator in the BK definition of cardinal numbers. (shrink)

Neofregeanism and structuralism are among the most promising recent approaches to the philosophy of mathematics. Yet both have serious costs. We develop a view, structuralist neologicism, which retains the central advantages of each while avoiding their more serious costs. The key to our approach is using arbitrary reference to explicate how mathematical terms, introduced by abstraction principles, refer. Focusing on numerical terms, this allows us to treat abstraction principles as implicit definitions determining all properties of the numbers, achieving a key (...) neofregean advantage, while preserving the key structuralist advantage, which objects play the number role does not matter. (shrink)

PG (Plural Grundgesetze) is a consistent second-order system which is aimed to derive second-order Peano arithmetic. It employs the notion of plural quantification and a few Fregean devices, among which the infamous Basic Law V. George Boolos’ plural semantics is replaced with Enrico Martino’s Acts of Choice Semantics (ACS), which is developed from the notion of arbitrary reference in mathematical reasoning. Also, substitutional quantification is exploited to interpret quantification into predicate position. ACS provides a form of logicism which is radically (...) alternative to Frege’s and which is grounded on the existence of individuals rather than on the existence of concepts. (shrink)

PLV (Plural Basic Law V) is a consistent second-order system which is aimed to derive second-order Peano arithmetic. It employs the notion of plural quantification and a first-order formulation of Frege's infamous Basic Law V. George Boolos' plural semantics is replaced with Enrico Martino's Acts of Choice Semantics (ACS), which is developed from the notion of arbitrary reference in mathematical reasoning. ACS provides a form of logicism which is radically alternative to Frege's and which is grounded on the existence of (...) individuals rather than on the existence of concepts. (shrink)

We reply to various arguments by Otavio Bueno and Edward Zalta (‘Object Theory and Modal Meinongianism’) against Modal Meinongianism, including that it presupposes, but cannot maintain, a unique denotation for names of fictional characters, and that it is not generalizable to higher-order objects. We individuate the crucial difference between Modal Meinongianism and Object Theory in the former’s resorting to an apparatus of worlds, possible and impossible, for the representational purposes for which the latter resorts to a distinction between two kinds (...) of predication, exemplification and encoding. We show that encoding has fewer forerunners in the history of philosophy than Bueno and Zalta want, and that there’s a reason why the notion has been found baffling by some. (shrink)

I would like to attack a certain view: The view that the concept of identity can fail to apply to some things although, for some positive integer n, we have n of them. The idea of entities without self-identity is seriously entertained in the philosophy of quantum mechanics. It is so pervasive that it has been labelled the Received View. I introduce the Received View in Section 1. In Section 2 I explain what I mean by entity, and I argue (...) that supporters of the Received View should agree with my characterization of the corresponding notion of entity. I also explain what I mean by identity, and I show that supporters of the Received View agree with my characterization of that notion. In Section 3 I argue that the concept of identity, so characterized, is one with the concept of oneness. Thus, it cannot but apply to what belongs to a collection with n elements, n being a positive integer. In Section 4 I add some considerations on the primitiveness of identity or unity and the status of the Identity of Indiscernibles. In Section 5 I address the problem of how reference to indiscernible objects with identity can be achieved. (shrink)

A version of the permutation argument in the philosophy of mathematics leads to the thesis that mathematical terms, contrary to appearances, are not genuine singular terms referring to individual objects; they are purely schematic or variables. By postulating ‘ante-rem structures’, the ante-rem structuralist aims to defuse the permutation argument and retain the referentiality of mathematical terms. This paper presents two semantic problems for the ante- rem view: (1) ante-rem structures are themselves subject to the permutation argument; (2) the ante-rem structuralist (...) fails to explain reference in a way that makes her account different to, and privileged over, that of her eliminativist rivals. Both problems undercut the motivation behind ante-rem structuralism. (shrink)

Cases of grounding failure present a puzzle for fundamental metaphysics. Typically, solutions are thought to lie either in adding ontology such as haecceities or in re‐describing the cases by means of the ideology of metaphysical indeterminacy. The controversial status of haecceities has led some to favour metaphysical indeterminacy as the way to solve the puzzle. We consider two further treatments of grounding failure each of which, we argue, is a more plausible alternative. As such, the initial dichotomy is a false (...) one, and these alternative options deserve consideration before resorting to the heavyweight machinery of metaphysical indeterminacy. -/- . (shrink)

It’s commonly held that particular moral facts are explained by ‘natural’ or ‘descriptive’ facts, though there’s disagreement over how such explanations work. We defend the view that general moral principles also play a role in explaining particular moral facts. More specifically, we argue that this view best makes sense of some intuitive data points, including the supervenience of the moral upon the natural. We consider two alternative accounts of the nature and structure of moral principles—’the nomic view’ and ‘moral platonism’—before (...) considering in what sense such principles obtain of necessity. (shrink)

Informal rigour is the process by which we come to understand particular mathematical structures and then manifest this rigour through axiomatisations. Structural relativity is the idea that the kinds of structures we isolate are dependent upon the logic we employ. We bring together these ideas by considering the level of informal rigour exhibited by our set-theoretic discourse, and argue that different foundational programmes should countenance different underlying logics (intermediate between first- and second-order) for formulating set theory. By bringing considerations of (...) perturbations in modal space to bear on the debate, we will suggest that a promising option for representing current set-theoretic thought is given by formulating set theory using quasi-weak second-order logic. These observations indicate that the usual division of structures into \particular (e.g. the natural number structure) and general (e.g. the group structure) is perhaps too coarse grained; we should also make a distinction between intentionally and unintentionally general structures. (shrink)

I argue that we can and should extend Tarski's model-theoretic criterion of logicality to cover indefinite expressions like Hilbert's ɛ operator, Russell's indefinite description operator η, and abstraction operators like 'the number of'. I draw on this extension to discuss the logical status of both abstraction operators and abstraction principles.

According to possibilism, or non-actualism, fictional characters are possible individuals. Possibilist accounts of fiction do not only assign the intuitively correct truth-conditions to sentences in a fiction, but has the potential to provide powerful explanatory models for a wide range of phenomena associated with fiction (though these two aspects of possibilism are, I argue, crucially distinct). Apart from the classic defense by David Lewis the idea of modeling fiction in terms of possible worlds have been widely criticized. In this article, (...) I provide a defense of a possibilist account against some lines of criticism. To do so, I assume that names for fictional characters are directly referential and a possible-worlds model that accommodates transworld identity. On this background, I argue, it is possible to construct an elegant model of fictional discourse using familiar models of information exchange in ordinary discourse, and I sketch how this model can be used to i) make a natural distinction between fictional and counterfactual discourse, ii) account for creativity, and iii) sustain a natural definition of truth-in-fiction that avoids certain familiar objections to possibilism. Though I set aside questions about the metaphysical commitments of a possible-world interpretation here, there is accordingly reason to think that the battle over possibilist treatments of fiction will have to be fought over metaphysical foundations rather than technical shortcomings. (shrink)

Existential instantiation is a rule of inference that allows us infer, from the proposition that there are some p things, the proposition that a is a p thing. What role does 'a' play here? According to one account, recently defended by Breckenridge and Magidor, we use 'a' to refer to a p thing. I argue that this cannot be right. I propose an alternative account, according to which we use 'a' to refer to a supposedly p thing.